If all menu settings are lost after migration, that typically means the migration tool incorrectly edits site data (e.g. the site URL) in the menu configuration and accidentally corrupts the configuration.
Enabling menu configuration compression may help prevent the configuration from being corrupted. To do that, go to the “Settings” tab, scroll down to “Debugging”, check the box that says “Compress menu configuration data that’s stored in the database”, and save the settings. Then go back to the “Admin Menu” and click “Save Changes” to store your existing menu settings in the new, compressed format.
By the way, in case you want to try manually copying the menu settings, they are stored in the wp_options table, in an option named ws_menu_editor.
